Expected Value

Probability Level 3

Choose seven numbers randomly and uniformly from the unit interval [ 0 , 1 ] [0,1] . What is the expected value of the third largest number of these seven numbers?


The answer is 0.625.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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2 solutions

Miles Koumouris
Dec 11, 2017

For some 0 ≤ x ≤ 1 0\leq x\leq 1 , the probability P ( x ) P(x) that x x is the third largest of the seven chosen numbers is given by P ( x ) = ( 6 4 ) x 4 ( 1 − x ) 2 P(x)=\binom64 x^4(1-x)^2 so that we can form the probability density function f ( x ) = 105 x 4 ( 1 − x ) 2 . f(x)=105x^4(1-x)^2. Then ∫ 0 1 105 x 5 ( 1 − x ) 2 d x = 0.625 . \int_0^1105x^5(1-x)^2\; dx=\boxed{0.625}.

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k/n+1 is the kth expected value of kth smallest number when choosing n numbers from 0 to 1

space sizzlers - 1 year, 2 months ago
Kelvin Hong
Dec 11, 2017

smallest to biggest 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.825 am I correct?

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